To find the antiderivative of $(x^2 + 2x + 5)^2$, we can proceed as follows:

Step 1: Expand the expression

First, expand $(x^2 + 2x + 5)^2$.

$(x^2 + 2x + 5)^2 = (x^2 + 2x + 5)(x^2 + 2x + 5)$

Using the distributive property (FOIL method), we get:

$(x^2 + 2x + 5)^2 = x^4 + 2x^3 + 5x^2 + 2x^3 + 4x^2 + 10x + 5x^2 + 10x + 25$

Now, combine like terms:

$x^4 + 4x^3 + 14x^2 + 20x + 25$

Step 2: Find the antiderivative

Now, integrate the expanded polynomial:

$\int (x^4 + 4x^3 + 14x^2 + 20x + 25) \, dx$

Use the power rule of integration:

$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$

Applying this to each term:

$\int x^4 \, dx = \frac{x^5}{5}, \quad \int 4x^3 \, dx = x^4, \quad \int 14x^2 \, dx = \frac{14x^3}{3}$ $\int 20x \, dx = 10x^2, \quad \int 25 \, dx = 25x$

Step 3: Combine the results

Now, combine all the terms:

$\int (x^2 + 2x + 5)^2 \, dx = \frac{x^5}{5} + x^4 + \frac{14x^3}{3} + 10x^2 + 25x + C$

where $C$ is the constant of integration.

Final Answer:

$\int (x^2 + 2x + 5)^2 \, dx = \frac{x^5}{5} + x^4 + \frac{14x^3}{3} + 10x^2 + 25x + C$

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Would you like further details on any part of this process? Or do you have any follow-up questions?

Here are 5 related questions you might find useful:

1. What is the power rule of integration?
2. How do you expand binomials like $(x^2 + 2x + 5)^2$?
3. How can you check your antiderivative by differentiation?
4. What is the role of the constant of integration $C$?
5. How do you solve more complex integrals involving polynomials?

Tip: When dealing with integrals of binomials raised to a power, always expand first before attempting to integrate. It simplifies the process!